Evening Bright | Smart Notes Library

Simple Harmonic Motion

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Simple harmonic motion (SHM) is a special type of oscillation where a particle moves back and forth on a straight line. The acceleration of the particle is always directed towards a fixed point (center of oscillation) and is proportional to the displacement from that point.

['a = − ω^2x', 'F = − kx', 'k = mω^2']

Harmonic/Periodic Motion: When a body repeats its motion after regular time intervals. The time interval is called the time period.
Oscillations: To and fro motion on the same path.
Simple Harmonic Motion (SHM): Oscillation on a straight line where the acceleration is always directed towards a fixed point and proportional to the displacement from this point (center of oscillation).
Defining Equation of SHM: a = -ω²x, where ω² is a positive constant and x is displacement.
If looking from an inertial frame, a = F/m. Thus, F = -mω²x = -kx
Force Constant/Spring Constant: k = mω².
The resultant force is zero at the center of oscillation (equilibrium position).
Restoring Force: A force that takes the particle back towards the equilibrium position. F = -kx represents a linear restoring force.

Simple Harmonic Motion Basics

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A block attached to a spring oscillates back and forth. The force exerted by the spring is proportional to the displacement from its natural length. The block's speed changes throughout the oscillation, being maximum at the center and zero at the extreme points. Energy is conserved in this system, and the maximum displacement is called the amplitude.

F = -kx

When a block attached to a spring is displaced from its natural length (position O) to a point P (displacement A) and released, it undergoes simple harmonic motion. The force on the particle is given by F = -kx, where k is the spring constant and x is the displacement from the natural length. The motion is simple harmonic because the resultant force is proportional to the displacement and acts towards the center (O). As the block moves from P to O, its speed increases, reaching a maximum at O. As it moves from O to Q (another extreme point), its speed decreases to zero. The potential energy at P and Q are equal due to conservation of energy, meaning OP = OQ. The block oscillates between P and Q, with the maximum displacement from the center (amplitude) being equal on both sides.

Equation of Motion of Simple Harmonic Motion

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The motion of a particle under a force proportional to displacement (F = -kx) is described. The equation of motion is derived using Newton's second law, leading to a differential equation that relates acceleration to displacement. This equation can be integrated to find the velocity as a function of displacement.

['F = -kx', 'a = -ω²x', 'ω = √(k/m)', 'vdv = − ω² x dx']

Consider a particle of mass m moving along the X-axis under the influence of a force F = -kx, where k is a positive constant and x is the displacement from the origin. This force results in simple harmonic motion with the center of oscillation at the origin. At t=0, the particle's position is x0 and its velocity is v0. The acceleration of the particle at any instant is: a = F/m = -kx/m = -ω²x, where ω = √(k/m). This leads to the differential equation: dv/dt = -ω²x Which can be rewritten as: v(dv/dx) = -ω²x Integrating both sides allows to find the relationship between velocity v and displacement x.

Phase Constant

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The phase constant (δ) determines the initial status of a particle in SHM and depends on the choice of t=0. Different choices of t=0 lead to different values of δ, but the motion remains the same. We often choose δ = 0 for simplicity.

x = A sin(ωt + δ)

The constant δ appearing in the equation x = A sin(ωt + δ) is called the phase constant.
The phase constant depends on the choice of the instant t=0.
Choosing t=0 at the mean position going towards positive direction makes δ = 0, leading to x = A sin(ωt).
Choosing t=0 at the positive extreme position makes δ = π/2, leading to x = A cos(ωt).
Any instant can be chosen as t=0, hence the phase constant can be chosen arbitrarily.
Often δ = 0 is chosen for simplicity.
General equation: x = A sin(ωt + δ) = A cos(ωt + δ'), where δ' is another arbitrary constant. Sine and cosine forms are equivalent, but the value of the phase constant depends on the chosen form.

Frequency and Angular Frequency

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Frequency (ν) is how many oscillations happen per unit time, measured in Hertz (Hz). Angular frequency (ω) is related to frequency and period (T).

ν = 1/T, ν = ω/2π, ω = 2πν, ω = √(k/m)

The reciprocal of the time period is called the frequency.
Frequency (ν) represents the number of oscillations per unit time.
Frequency is measured in cycles per second (Hertz, Hz).
ω = 2πν = 2π/T
ν = 1/T = ω/2π = (1/2π) * √(k/m)
ω = √(k/m)
The constant ω is called the angular frequency.

Phase

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Phase (Φ = ωt + δ) tells you where a particle is in its SHM cycle. Adding 2π to the phase brings the particle back to the same status.

φ = ωt + δ

The quantity φ = ωt + δ is called the phase.
It determines the status of the particle in simple harmonic motion.
If the phase is zero, the particle is crossing the mean position and going towards the positive direction.
If the phase is π/2, the particle is at the positive extreme position.
A phase ωt + δ is equivalent to a phase ωt + δ + 2π (or 4π, 6π, etc.).
As time increases, the phase increases.

Simple Harmonic Motion as a Projection of Circular Motion

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Simple harmonic motion can be understood as the projection of uniform circular motion onto a diameter of the circle.

['x = A cos ωt', 'y = A sin ωt']

Consider a particle P moving on a circle of radius A with constant angular speed ω. The x and y coordinates of the particle at time t are given by: x = A cos ωt y = A sin ωt These equations represent simple harmonic motions along the X and Y axes respectively, with amplitude A and angular frequency ω. The phase difference between the two SHMs is π/2.

Energy Conservation in Simple Harmonic Motion

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In simple harmonic motion, the total mechanical energy (sum of potential and kinetic energy) remains constant.

['F = -kx', 'dW = -kx dx', 'W = -1/2 kx^2', 'U(x) = 1/2 kx^2', 'k = mω^2', 'U(x) = 1/2 mω^2 x^2', 'U = 1/2 mω^2 A^2 sin^2(ωt + δ)', 'K = 1/2 m A^2 ω^2 cos^2(ωt + δ)', 'E = 1/2 mω^2 A^2']

Simple harmonic motion is defined by the equation F = -kx. The work done by the force F during a displacement from x to x + dx is dW = -kx dx. The work done in a displacement from x = 0 to x is W = -1/2 kx^2. The potential energy U(x) is 1/2 kx^2, chosen to be zero at x=0. Since ω = sqrt(k/m), k = mω^2 and thus U(x) = 1/2 mω^2 x^2. Given x = A sin(ωt + δ) and v = Aω cos(ωt + δ), the potential energy at time t is U = 1/2 mω^2 A^2 sin^2(ωt + δ) and the kinetic energy at time t is K = 1/2 m A^2 ω^2 cos^2(ωt + δ). The total mechanical energy at time t is E = U + K = 1/2 mω^2 A^2, which is constant.

Energy in SHM

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The total energy in SHM is the sum of potential and kinetic energies, and it remains constant if no energy is lost to friction, etc.

['U = (1/2) * k * θ^2 = (1/2) * I * ω^2 * θ^2', 'K = (1/2) * I * Ω^2', 'E = U + K = (1/2) * I * ω^2 * θ_0^2']

Let OQ be the horizontal line in the plane of motion. Let PQ be the perpendicular to OQ. The potential energy is U = (1/2) * k * θ^2 = (1/2) * I * ω^2 * θ^2 and the kinetic energy is K = (1/2) * I * Ω^2 The total energy is E = U + K = (1/2) * I * ω^2 * θ_0^2 * sin^2(ωt + δ) + (1/2) * I * θ_0^2 * ω^2 * cos^2(ωt + δ) = (1/2) * I * ω^2 * θ_0^2

Simple Pendulum

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A simple pendulum consists of a point mass suspended by a massless, inextensible string. For small angular displacements, its motion approximates SHM.

['Γ = -mgl sinθ', 'For small angles: Γ = -mgl θ', 'I = ml^2', 'α = - (g/l) * θ = -ω^2 * θ', 'ω = √(g/l)', 'T = 2π√(l/g)']

A simple pendulum consists of a heavy particle suspended from a fixed support through a light inextensible string. Simple pendulum is an idealised model. In practice, one takes a small metallic sphere and suspends it through a string. Forces acting on the particle are (a) the weight mg and (b) the tension T. The torque of T about OA is zero as it intersects OA. The magnitude of the torque of mg about OA is |Γ| = (mg)(OQ) = mg(OP)sinθ = mgl sinθ. Also, the torque tries to bring the particle back towards θ = 0. Thus, we can write Γ = − mgl sinθ. We see that the resultant torque is not proportional to the angular displacement and hence the motion is not angular simple harmonic. However, if the angular displacement is small, sinθ is approximately equal to θ (expressed in radians) and equation (12.21) may be written as Γ = − mgl θ. Thus, if the amplitude of oscillation is small, the motion of the particle is approximately angular simple harmonic. The moment of inertia of the particle about the axis of rotation OA is I = m(OP)^2 = ml^2. The angular acceleration is α = Γ/I = −(mgl θ) / (ml^2) = −(g/l) θ or α = − ω^2 θ where ω = √(g/l). This is the equation of an angular simple harmonic motion. The constant ω = √(g/l) represents the angular frequency. The time period is T = 2π/ω = 2π √(l/g).

Simple Pendulum

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A simple pendulum's motion can be approximated as simple harmonic motion (SHM) for small oscillations. Its time period depends on the length of the pendulum and the acceleration due to gravity.

['T = 2π √(l/g)', 'ω = √(g/l)', 'd²x/dt² = -ω²x', 'g = 4π²l / T²']

The time period of a simple pendulum is given by T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.
For small oscillations, the motion is approximately linear SHM.
The angular frequency ω = √(g/l).
The equation of motion is d²x/dt² = -ω²x.
The restoring force is proportional to the displacement from the equilibrium position.
Experimentally, the value of 'g' can be determined by measuring the time period for a known length of the pendulum.

Physical Pendulum

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A rigid body suspended from a fixed point constitutes a physical pendulum. For small oscillations, its motion approximates simple harmonic motion.

T = 2π√(I / mgl), where I is the moment of inertia about the pivot point, m is the mass, g is the acceleration due to gravity, and l is the distance from the pivot point to the center of mass.

Any rigid body suspended from a fixed support constitutes a physical pendulum. Examples include a circular ring suspended on a nail or a heavy metallic rod suspended through a hole.

When the center of mass C is vertically below O, the body is at rest (θ = 0). When displaced and released, it oscillates.

The body rotates about a horizontal axis through O (axis OA). At time t, the angular displacement is θ.

Forces acting on the body: (a) weight mg, (b) contact force N at O.

Torque of N about OA is zero. Torque of mg is |Γ| = mg(OD) = mg(OC)sinθ = mgl sinθ, where l = OC (separation between suspension point and center of mass). This torque restores the body towards θ = 0. Thus, Γ = -mglsinθ.

If I is the moment of inertia about OA, the angular acceleration is α = Γ/I = -(mgl/I)sinθ.

For small displacements, sinθ ≈ θ, so α ≈ -(mgl/I)θ. Thus, for small oscillations, the motion is nearly simple harmonic.

Torsional Pendulum

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A torsional pendulum involves an extended body suspended by a wire. Twisting the body creates a restoring torque proportional to the angle of twist, resulting in oscillations.

Γ = -kθ, where k is the torsional constant and θ is the angle of twist. α = - (k/I) θ, where I is the moment of inertia.

In a torsional pendulum, an extended body is suspended by a light thread or wire.

The body is rotated through an angle about the wire (axis of rotation). This produces a twist in the wire.

The twisted wire exerts a restoring torque on the body, bringing it back to its original position. The torque's magnitude is proportional to the angle of twist (θ).

The proportionality constant is the torsional constant (k) of the wire. Thus, the torque is Γ = -kθ.

If I is the moment of inertia about the vertical axis, the angular acceleration is α = Γ/I = -(k/I)θ

Combination of Simple Harmonic Motions

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When two simple harmonic motions with the same frequency occur along the same direction, their combination results in another simple harmonic motion. The amplitude and phase of the resultant motion depend on the amplitudes and phase difference of the individual motions.

['Resultant displacement: x = A sin(ωt + ε)', 'Resultant Amplitude: A = √(C^2 + D^2)', 'Where C = A1 + A2 cosδ and D = A2 sinδ', 'Resultant Amplitude (Alternative): A = √(A1^2 + A2^2 + 2 A1 A2 cos δ)', 'Phase Constant: tan ε = (A2 sin δ) / (A1 + A2 cos δ)']

When combining two SHM's: x = C sin(ωt) + D cos(ωt) = √(C^2 + D^2) [ (C/√(C^2 + D^2)) sin(ωt) + (D/√(C^2 + D^2)) cos(ωt) ] = A sin(ωt + ε) where A = √(C^2 + D^2). If x1 = A1 sin(ωt) and x2 = A2 sin(ωt + δ), then the resultant amplitude A = √(A1^2 + A2^2 + 2A1A2 cos δ). The resultant motion's phase is given by tan ε = (A2 sin δ) / (A1 + A2 cos δ)

Composition of Simple Harmonic Motions

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When multiple SHMs act on a particle, the resultant motion depends on the directions, amplitudes, frequencies, and phases of the individual SHMs. The motion can be a straight line, an ellipse, or more complex, based on the specific conditions.

['Resultant Amplitude (A) for two SHMs in the same direction: A = sqrt[ A1^2 + A2^2 + 2A1A2 cos(delta) ]', 'tan(epsilon) = (A2 sin(delta)) / (A1 + A2 cos(delta))', 'Equation of Elliptical Path: (x^2 / A1^2) + (y^2 / A2^2) - (2xy cos(delta) / A1A2) = sin^2(delta)']

Composition of Simple Harmonic Motions

(A) Addition of Simple Harmonic Motions in the Same Direction

When adding multiple SHMs in the same direction, the resultant motion is also a SHM.

(B) Composition of Two Simple Harmonic Motions in Perpendicular Directions

Suppose two forces act on a particle, the first alone would produce a simple harmonic motion in the x-direction given by x = A₁ sinωt ... (i) and the second would produce a simple harmonic motion in the y-direction given by y = A₂ sin(ωt + δ) ... (ii) The amplitudes A₁ and A₂ may be different and their phases differ by δ. The frequencies of the two simple harmonic motions are assumed to be equal. The resultant motion of the particle is a combination of the two simple harmonic motions. The position of the particle at time t is (x, y) where x is given by equation (i) and y is given by (ii). The motion is thus two-dimensional and the path of the particle is in general an ellipse. The equation of the path may be obtained by eliminating t from (i) and (ii).

Equation of the ellipse:

(x² / A₁²) + (y² / A₂²) - (2xy cosδ / A₁A₂) = sin²δ ... (12.29)

Special Cases

(a) δ = 0: The two simple harmonic motions are in phase.

y = (A₂ / A₁) x ... (iii)

Superposition of Two Simple Harmonic Motions

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When two SHMs occur simultaneously, the resulting motion depends on their amplitudes and phase difference. Specific cases include straight-line motion when the phase difference is 0 or π, and elliptical or circular motion when the phase difference is π/2.

['Resultant displacement when δ=0: r = √(A₁² + A₂²) sin(ωt)', 'Resultant displacement when δ=π: y = -(A₂/A₁)x', 'Equation of ellipse when δ=π/2: x²/A₁² + y²/A₂² = 1', 'Equation of circle when A₁ = A₂ = A and δ=π/2: x² + y² = A²']

The text discusses the superposition of two simple harmonic motions along perpendicular directions. It examines cases with different phase differences (δ) and derives equations for the resultant motion. **Case (a): δ = 0** The equation of the path is a straight line passing through the origin with a slope of tan⁻¹(A₂/A₁). The displacement is given by r = √(A₁² + A₂²) sin(ωt), indicating simple harmonic motion with amplitude √(A₁² + A₂²). **Case (b): δ = π** The equation of the path is a straight line, y = -(A₂/A₁)x. The particle oscillates on this line. The displacement is given by r = √(A₁² + A₂²) sin(ωt), indicating simple harmonic motion with amplitude √(A₁² + A₂²). **Case (c): δ = π/2** The equations of motion are x = A₁sin(ωt) and y = A₂cos(ωt). The path is an ellipse inscribed in a rectangle. The general equation is x²/A₁² + y²/A₂² = 1, which is the standard equation of an ellipse with its axes along X and Y-axes and with its center at the origin. If A₁ = A₂ = A, the path becomes a circle, represented by x² + y² = A². Thus, two SHMs of equal amplitude in perpendicular directions with a phase difference of π/2 result in circular motion.

Forced Vibration and Resonance

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In forced vibration, if damping is zero, the amplitude at resonance would theoretically be infinite. However, some damping is always present, keeping the amplitude finite but potentially very large if the driving frequency is close to the natural frequency. This is crucial in designing structures like bridges to avoid catastrophic resonance effects.

Amplitude at resonance (ideally, with zero damping) → ∞

Forced Vibration: Vibration caused by an external periodic force.
Resonance: A condition where the driving frequency of the external force matches the natural frequency of the system, leading to a large amplitude vibration.
Damping: A phenomenon that dissipates energy from the oscillating system, reducing the amplitude of vibration.
Ideally, without damping, the amplitude at resonance would be infinite according to equation (12.31).
In reality, some damping is always present, which keeps the amplitude finite.
Small damping and driving frequency close to natural frequency can result in large amplitude.
This effect is important in civil engineering designs, especially for bridges, to avoid resonance-induced failures.

Simple Harmonic Motion Equations

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SHM can be described mathematically. The displacement, velocity, and acceleration of a particle in SHM can be calculated using trigonometric functions. Key parameters include amplitude, angular frequency, and phase.

General Equation: x = A sin(ωt + δ), where A is the amplitude, ω is the angular frequency, t is time, and δ is the phase constant.
Amplitude (A): The maximum displacement from the mean position.
Time Period (T): The time taken for one complete oscillation, T = 2π/ω.
Maximum Speed: v_max = Aω.
Maximum Acceleration: a_max = Aω².
Velocity at a given displacement: v = ω√(A² - x²).

Vertical Oscillations of a Spring-Mass System

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When a mass is attached to a vertical spring and oscillates, gravity only shifts the equilibrium position. The time period of oscillation is independent of gravity and depends only on the mass and spring constant.

['Equilibrium extension: ∆l = mg/k', 'Time period: T = 2π√(m/k)', 'Angular frequency: ω = √(k/m)', 'Frequency: ν = ω / (2π)', 'Maximum speed: v_max = Aω', 'Speed at a given position: v = ω√(A² - x²)']

The equilibrium position is shifted by a distance mg/k when a mass m is attached to a spring with spring constant k.
The time period of oscillation is given by T = 2π√(m/k).
Gravity does not affect the time period but only shifts the equilibrium.

Simple Harmonic Motion

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Explains concepts related to SHM including energy, time period calculations for different systems involving pulleys, springs and blocks, and the effect of friction and collisions on SHM.

['U = 1/2 (I/r^2 + m)v^2 + m^2g^2/(8k) + 1/2 kx^2', 'ω^2 = 4k / (I/r^2 + m)', 'T = 2π sqrt((I/r^2 + m) / (4k))', 'A = µ(M + m)g / k', 'A = v * sqrt(m/(2k))', 'T = π * sqrt(m/k) + 2L/v']

The potential energy U of a system in SHM is described, and the conservation of energy is used to derive the equation of motion and the angular frequency ω.
The time period T for a system involving a pulley and a spring is derived.
The effect of friction on the amplitude of SHM is discussed.
The effect of inelastic collisions on the amplitude of resulting SHM is explained.
Motion of a block connected to a spring and colliding elastically with walls is described.

SHM of a Block in an Elevator After Cable Breaks

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When the cable of a stationary elevator breaks, the block attached to a spring inside the elevator executes SHM. The mean position corresponds to the unstretched spring, and the amplitude is determined by the initial stretch of the spring due to the block's weight.

Amplitude = mg/k

When the elevator is stationary, the spring is stretched to support the block. If the extension is x, the tension is kx which should balance the weight of M + m the block. Thus, x = mg/k. As the cable breaks, the elevator starts 0 falling with acceleration ‘g’. We shall work in the frame of reference of the elevator. Then we have to use a pseudo force mg upward on the block. This force will ‘balance’ the weight. Thus, the block is subjected to a net force kx by the spring when it is at a distance x from the position of unstretched spring. Hence, its motion in the elevator is simple harmonic with its mean position corresponding to the unstretched spring. Initially, the spring is stretched by x = mg/k, where the velocity of the block (with respect to the elevator) is zero. Thus, the amplitude of the resulting simple harmonic motion is mg/k.

Frequency of Oscillation of Two Blocks Connected by a Spring

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Two blocks connected by a spring oscillate with a frequency determined by the spring constant and the masses of the blocks. The frequency depends on the effective mass (reduced mass) of the system.

['Angular frequency: ω = sqrt(k(M+m)/Mm)', 'Frequency: ν = (1/2π) * sqrt(k(M+m)/Mm)']

Considering “the two blocks plus the spring” as a system, there is no external resultant force on the system. Hence the centre of mass of the system will remain at rest. The mean positions of the two simple harmonic motions occur when the spring becomes unstretched. If the mass m moves towards right through a distance x and the mass M moves towards left through a distance X before the spring acquires natural length, x + X = x. … (i) x and X will be the amplitudes of the two blocks m and 0 M respectively. As the centre of mass should not change during the motion, we should also have mx = MX. … (ii) From (i) and (ii), x = Mx 0 and X = mx 0 ⋅ M + m M + m Hence, the left block is x = 0 distance away from its M + m position in the beginning of the motion. The force by the spring on this block at this instant is equal to the tension of spring, i.e., T = kx. Now x = 0 or, x = x Mx M + m k(M + m) T k(M + m) or, a = = x. M m Mm The angular frequency is, therefore, ω = √ k(M + m) Mm ω 1 k(M + m) and the frequency is ν = = ⋅ 2π 2π Mm

SHM in a Tunnel Through the Earth

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A particle released in a tunnel dug through the Earth executes SHM. The restoring force is proportional to the distance from the Earth's center, and the time period depends on Earth's density and gravitational constant.

['Force constant: k = GMm/R^3', 'Time period: T = 2π * sqrt(R^3/GM)']

Consider the situation shown in figure (12-W9). Suppose at an instant t the particle in the tunnel is at a distance x from the centre of the earth. Let us draw a sphere of radius x with its centre at the centre of the earth. Only the part of the earth within this sphere will exert a net attraction on the particle. Mass of this part is M′ = 4 π x 3 M = x 3 M. 3 4 π R 3 R 3 The force of attraction is, therefore, F = G(x 3/R 3) Mm = GMm x. x 2 R 3 This force acts towards the centre of the earth. Thus, the resultant force on the particle is opposite to the displacement from the centre of the earth and is proportional to it. The particle, therefore, executes a simple harmonic motion in the tunnel with the centre of the earth as the mean position. The force constant is k = GMm , so that the time period is R 3 T = 2π √ = 2π √R 3⋅ M k GM

Physical Pendulum

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A physical pendulum is an extended object that oscillates about a pivot point. Its time period depends on the moment of inertia about the pivot, the mass, and the distance from the pivot to the center of mass.

T = 2π √(I/(mgd))

The time period of a physical pendulum is given by T = 2π sqrt(I/(mgd)), where I is the moment of inertia about the point of suspension, m is the mass, g is the acceleration due to gravity, and d is the distance between the center of mass and the point of suspension.

Torsional Pendulum

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A torsional pendulum consists of an object suspended by a wire that twists. Its time period depends on the moment of inertia of the object and the torsional constant of the wire.

T = 2π √(I/K)

The time period of a torsional pendulum is given by T = 2π sqrt(I/K), where I is the moment of inertia about the suspension wire and K is the torsional constant of the wire.

Effective g

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In a non-inertial frame, we can use the concept of effective 'g' to analyze SHM. This effective 'g' accounts for pseudo forces.

geff = √(g^2 + a0^2)

When a system experiences an acceleration a0, the effective 'g' can be calculated as sqrt(g^2 + a0^2). This value is then used to calculate time periods of SHM.

Resultant of two SHM's

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When a particle is subjected to two SHMs of the same frequency, the resultant motion is also a SHM with the same frequency. The amplitude and phase of the resultant motion depend on the amplitudes and phase difference of the individual SHMs.

A = √⎯⎯ A ⎯ 2 ⎯+⎯ A ⎯ 2 ⎯ +⎯ 2 ⎯ ⎯ A ⎯ A ⎯⎯ c ⎯ o ⎯ s(δ) v_max = Aω a_max = Aω^2

The resultant of two simple harmonic motions is a simple harmonic motion of the same angular frequency ω. The amplitude of the resultant motion is A = √⎯⎯ A ⎯ 2 ⎯+⎯ A ⎯ 2 ⎯ +⎯ 2 ⎯ ⎯ A ⎯ A ⎯⎯ c ⎯ o ⎯ s(⎯π⎯ / ⎯ 3)

Simple Harmonic Motion

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This section covers the fundamental concepts and properties of simple harmonic motion (SHM), including displacement, velocity, acceleration, time period, and energy.

['x = A cos(ωt + φ)', 'v = -Aωsin(ωt + φ)', 'a = -ω^2x', 'T = 2π/ω', 'f = ω/2π', 'U = (1/2)kx^2 = (1/2)mω^2x^2', 'E = (1/2)kA^2 = (1/2)mω^2A^2']

Simple harmonic motion is a special type of periodic motion where the restoring force is directly proportional to the displacement, causing the object to oscillate about an equilibrium position.
The equation of motion for SHM can be expressed as x = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase constant.
Velocity in SHM: v = -Aωsin(ωt + φ)
Acceleration in SHM: a = -Aω^2cos(ωt + φ) = -ω^2x
Time period (T) and frequency (f) are related to the angular frequency by T = 2π/ω and f = ω/2π.
The potential energy (U) in SHM varies with displacement: U = (1/2)kx^2 = (1/2)mω^2x^2
The kinetic energy (K) also varies with time and position.
The total mechanical energy (E) in SHM remains constant: E = (1/2)kA^2 = (1/2)mω^2A^2

Simple Harmonic Motion

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Simple Harmonic Motion (SHM) is a periodic and oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

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Simple Harmonic Motion (SHM): A special type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position and acts towards the equilibrium.
Periodic Motion: Motion that repeats itself after a fixed interval of time.
Oscillatory Motion: Motion that moves back and forth around an equilibrium position.

Total Mechanical Energy in SHM

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The total mechanical energy of a spring-mass system in SHM is given by E = (1/2) * m * ω^2 * A^2, where m is the mass, ω is the angular frequency, and A is the amplitude.

['E = (1/2) * m * ω^2 * A^2']

Total Mechanical Energy (E): The sum of kinetic and potential energies in SHM, which remains constant if there are no dissipative forces.
Formula: E = (1/2) * m * ω^2 * A^2
Where:
- m = mass
- ω = angular frequency
- A = amplitude

Simple Harmonic Motion

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Simple harmonic motion (SHM) is a periodic motion where the restoring force is directly proportional to the displacement, causing oscillation around an equilibrium position.

Simple Harmonic Motion (SHM): A special type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position and acts in the opposite direction.
Displacement (x): The distance of the particle from its mean position.
Amplitude (A): The maximum displacement from the mean position.
Time Period (T): The time taken to complete one full oscillation.
Frequency (f): The number of oscillations per unit time (f = 1/T).
Angular Frequency (ω): ω = 2πf = 2π/T
Equation of SHM: x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + φ), where φ is the phase constant.
Velocity in SHM: v(t) = dx/dt = -Aω sin(ωt + φ) or v(t) = Aω cos(ωt + φ) (depending on the initial equation).
Maximum Velocity: v_max = Aω
Acceleration in SHM: a(t) = dv/dt = -Aω² cos(ωt + φ) = -ω²x(t)
Maximum Acceleration: a_max = Aω²
Kinetic Energy (KE) in SHM: KE = (1/2)mv²
Potential Energy (PE) in SHM: PE = (1/2)kx² = (1/2)mω²x² (where k is the spring constant)
Total Energy (E) in SHM: E = KE + PE = (1/2)kA² = (1/2)mω²A²
Average Potential Energy: In SHM, the average potential energy over one time period is equal to the average kinetic energy over the same time period.

Potential Energy of a Spring

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A stretched or compressed spring stores potential energy.

['Potential Energy of a Spring: U = (1/2)kx^2, where k is the spring constant and x is the displacement from equilibrium.']

When a spring is stretched or compressed, it stores potential energy due to the work done against its elastic force. This potential energy can be converted into kinetic energy when the spring is released.

Time Period of Oscillation

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The time period is the time it takes for one complete oscillation.

['Time Period (T) = 2π√(m/k), where m is the mass and k is the spring constant.', 'For springs in series: 1/keq = 1/k1 + 1/k2 + ...', 'For springs in parallel: keq = k1 + k2 + ...']

The time period of oscillation for a mass-spring system depends on the mass and the spring constant. Different arrangements of springs (series or parallel) result in different equivalent spring constants, affecting the time period.

Equilibrium Position

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Equilibrium is where the net force on the object is zero.

['At Equilibrium: Net Force = 0']

The equilibrium position is the point where the restoring force of the spring balances any other forces acting on the object, such as gravity. At equilibrium, the spring may be compressed or stretched depending on the situation.

Time Period of SHM

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This section deals with finding the time period of various systems undergoing simple harmonic motion, including blocks connected to springs, pendulums, and combinations thereof. The problems involve using concepts of elasticity, friction, and moment of inertia.

['Time period of a simple pendulum: T = 2π√(L/g)', 'Time period of a spring-mass system: T = 2π√(m/k)', 'Effective spring constant for springs in series and parallel (if applicable).', 'Formulas for elastic collisions (conservation of momentum and kinetic energy).']

The problems cover various scenarios of SHM. Key concepts involved are: * Elastic collisions * SHM of spring-mass systems (horizontal and vertical) * SHM of systems with pulleys * SHM of a rectangular plate suspended by strings * Effect of additional mass on the SHM of a spring-mass system * SHM on surfaces with friction * Simple Pendulum

Simple Pendulum

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A simple pendulum consists of a point mass (bob) suspended by a light, inextensible string. It oscillates with a time period that depends on the length of the string and the acceleration due to gravity. For small oscillations, the motion is approximately simple harmonic.

['T = 2π√(l/g)']

The time period of a simple pendulum undergoing small oscillations is given by: T = 2π√(l/g) where: * T is the time period * l is the length of the pendulum * g is the acceleration due to gravity

Oscillations in Different Scenarios

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The time period of oscillations can change when the pendulum is placed in accelerating frames or when it's not a simple pendulum (e.g., a physical pendulum). The effective 'g' changes in accelerating frames, and moment of inertia plays a role in physical pendulums.

['T = 2π√(l/(g + a₀)) (elevator accelerating upwards)', 'T = 2π√(l/(g - a₀)) (elevator accelerating downwards)', 'T = 2π√(l/g) (elevator moving with uniform velocity)', 'T = 2π√(I/mgd) (physical pendulum)', 'τ = -kθ (torsional oscillations)', 'T = 2π√(I/k) (torsional oscillations time period)']

1. **Pendulum in an Accelerating Frame:** * If the elevator is accelerating upwards with acceleration a₀, the effective acceleration due to gravity becomes g + a₀. * If the elevator is accelerating downwards with acceleration a₀, the effective acceleration due to gravity becomes g - a₀. * If the elevator is moving with uniform velocity, the effective acceleration due to gravity remains g. * The time period is adjusted accordingly in each case, T = 2π√(l/g_effective). 2. **Physical Pendulum:** * For a physical pendulum (an extended object oscillating about a pivot), the time period is given by: T = 2π√(I/mgd) where: * I is the moment of inertia about the pivot point * m is the mass of the object * g is the acceleration due to gravity * d is the distance from the pivot point to the center of mass 3. **Torsional Oscillations** * τ = -kθ (torque is proportional to angular displacement) * T = 2π√(I/k) where I is the moment of inertia and k is the torsional constant.

SHM Concepts

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This section deals with concepts and problems related to Simple Harmonic Motion, including oscillations, restoring forces, time period, amplitude, and phase differences.

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Simple Harmonic Motion (SHM): A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
Amplitude (A): The maximum displacement of the particle from its mean position.
Time Period (T): The time taken to complete one full oscillation.
Phase Difference: The difference in phase between two SHMs. It determines the relative positions and velocities of the oscillating objects at any given time.

Time Period Formulas

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Various formulas for calculating the time period of SHM in different scenarios are provided.

['T = 2π√(M(k1 + k2 + k3) / (k1k2 + k2k3 + k3k1))', 'T = 2π√(m/k)', 'T = 2π√(L/(g + a0))', 'T = 2π√(L/(g - a0))', 'T = 2π√(m/(2k))', 'T = 2π√(L/g)', 'T = 2π√((m + I/r^2)/k)', 'T = 2π√(R/g)', 'T = 2π√(R^3/(GM))', 'T = 2π√(2r/g)', 'T = 2π√(√8 a / g)', 'T = 2π√(3r/g)', 'T = 2π√((m^2 * L)/(k * v))', 'f = (5/2π) Hz']

The provided text presents a collection of formulas for calculating the time period of simple harmonic motion (SHM) in various systems. These formulas take into account factors such as spring constants, masses, lengths, gravitational acceleration, and other relevant physical parameters. Each formula applies to a specific configuration of oscillating system.

Angular Displacement and SHM

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The text offers formulas related to angular displacement and its connection to simple harmonic motion, along with calculations involving potential energy.

['T = 2π√(2mr^2 / (μg))', 'Potential Energy = (1/2)k0^2 θ^2 + (m^2 * g^2) / L^2']

These formulas relate to angular displacement in SHM problems. They often involve small angle approximations, potential energy calculations, and the determination of equilibrium positions.

Oscillations

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A type of periodic motion where the restoring force is proportional to the displacement.

Frequency: f= 1/T; Angular Frequency: ω = 2πf; Displacement: x(t) = Acos(ωt + φ)

Includes amplitude, frequency, time period, and different types of oscillations (free, forced).

SHM

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A type of periodic motion where the restoring force is proportional to the displacement.

F = -kx

Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

Amplitude

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The maximum displacement from the equilibrium position.

N/A

Amplitude is the maximum displacement of an object from its equilibrium position during SHM.

Angular Frequency

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The rate of change of the phase of a sinusoidal waveform.

ω = 2πf = 2π/T

Angular frequency is a measure of how quickly oscillations occur, expressed in radians per second.

Frequency

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The number of oscillations per unit time.

f = 1/T

Frequency is the number of complete oscillations or cycles that occur per unit of time. It is typically measured in Hertz (Hz).

Phase

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The position of a point in time (an instant) on a waveform cycle.

N/A

Phase refers to the position of a point in time (an instant) on a waveform cycle. It is a relative measurement that represents the fraction of the cycle that has elapsed relative to a reference point.

Time Period

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The time taken for one complete oscillation.

T = 1/f

Time period is the time taken for one complete oscillation or cycle in SHM.

Damped SHM

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SHM where the amplitude decreases over time due to energy loss.

N/A

Damped SHM is a type of SHM in which the amplitude of the oscillations decreases over time due to energy dissipation, typically due to friction or air resistance.

Simple Pendulum

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A mass suspended from a pivot point that swings back and forth.

T = 2π√(L/g)

A simple pendulum consists of a mass (bob) suspended from a pivot point by a light string or rod. It exhibits SHM for small angles of displacement.